Analysis Roulette Speed-Of-Play Study
analysis done by Henrik Singmann (singmann@gmail.com)
01 November, 2021
Preparation
Total participants (NULL = did not pass captchas):
##
## delay normal NULL
## 502 502 182Participants that passed the captchas and finished the study:
##
## Normal Slowed-down
## 500 502Descriptive Statistics
Note that the descriptives do not need to match the results from the models below.
Age
## # A tibble: 1 × 6
## age_min age_sd age_mean age_max age_NAs age_prop_NA
## <dbl> <dbl> <dbl> <dbl> <int> <dbl>
## 1 18 10.4 34.0 77 9 0.00898## # A tibble: 2 × 7
## exp_cond age_min age_sd age_mean age_max age_NAs age_prop_NA
## <fct> <dbl> <dbl> <dbl> <dbl> <int> <dbl>
## 1 Normal 18 9.82 33.6 77 6 0.012
## 2 Slowed-down 18 10.9 34.4 74 3 0.00598Gender
## # A tibble: 3 × 3
## Sex n prop
## <chr> <int> <dbl>
## 1 Female 340 0.339
## 2 Male 661 0.660
## 3 <NA> 1 0.000998## # A tibble: 5 × 4
## # Groups: exp_cond [2]
## exp_cond Sex n prop
## <fct> <chr> <int> <dbl>
## 1 Normal Female 167 0.334
## 2 Normal Male 332 0.664
## 3 Normal <NA> 1 0.002
## 4 Slowed-down Female 173 0.345
## 5 Slowed-down Male 329 0.655Payed bonus
Overall bonus paid out was:
## [1] 4.06If we split bonus by whether or not participants played, we see that playing leads to a slightly higher payout:
## # A tibble: 2 × 6
## bet_at_all mean_bonus sd_bonus se_bonus max_bonus min_bonus
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 bet 4.08 3.89 0.137 72 0
## 2 no bet 4 0 0 4 4PGSI Scores
Percentages problem gamblers:
## # A tibble: 1 × 3
## mean sd perc_larger_7
## <dbl> <dbl> <dbl>
## 1 3.09 3.90 0.116Percentages problem gamblers per condition:
## # A tibble: 2 × 4
## exp_cond mean sd perc_larger_7
## <fct> <dbl> <dbl> <dbl>
## 1 Normal 3.03 3.89 0.11
## 2 Slowed-down 3.15 3.91 0.122PGSI distribution Ferris & Wynne:
## # A tibble: 4 × 3
## bin n prob
## <fct> <int> <dbl>
## 1 no risk (0) 327 0.326
## 2 low risk (1 - 2) 259 0.258
## 3 moderate risk (3 - 7) 300 0.299
## 4 problem gambler (> 7) 116 0.116PGSI distribution Ferris & Wynne per condition:
## # A tibble: 8 × 4
## # Groups: exp_cond [2]
## exp_cond bin n prob
## <fct> <fct> <int> <dbl>
## 1 Normal no risk (0) 158 0.316
## 2 Normal low risk (1 - 2) 134 0.268
## 3 Normal moderate risk (3 - 7) 153 0.306
## 4 Normal problem gambler (> 7) 55 0.11
## 5 Slowed-down no risk (0) 169 0.337
## 6 Slowed-down low risk (1 - 2) 125 0.249
## 7 Slowed-down moderate risk (3 - 7) 147 0.293
## 8 Slowed-down problem gambler (> 7) 61 0.122PGSI distribution Currie et al.:
## # A tibble: 4 × 3
## bin n prob
## <fct> <int> <dbl>
## 1 recreational (0) 327 0.326
## 2 low risk (1 - 4) 424 0.423
## 3 moderate risk (5 - 7) 135 0.135
## 4 problem gambler (> 7) 116 0.116PGSI distribution Currie et al. per condition:
## # A tibble: 8 × 4
## # Groups: exp_cond [2]
## exp_cond bin n prob
## <fct> <fct> <int> <dbl>
## 1 Normal recreational (0) 158 0.316
## 2 Normal low risk (1 - 4) 227 0.454
## 3 Normal moderate risk (5 - 7) 60 0.12
## 4 Normal problem gambler (> 7) 55 0.11
## 5 Slowed-down recreational (0) 169 0.337
## 6 Slowed-down low risk (1 - 4) 197 0.392
## 7 Slowed-down moderate risk (5 - 7) 75 0.149
## 8 Slowed-down problem gambler (> 7) 61 0.122Distribution of Betting Times
We considered the time participants took for each round of roulette starting with the second round (as this was the first round to which the 1 minute slow down applied).
The first graph shows the distribution of individual betting times in seconds when restricting the shown range.
However, this plot excludes a few very long betting times, most of which occurred in the slowed down version.
## # A tibble: 19 × 3
## ppt_id exp_cond time
## <chr> <fct> <dbl>
## 1 P0842 Slowed-down 339
## 2 P0537 Slowed-down 348
## 3 P1016 Slowed-down 306
## 4 P0537 Slowed-down 1322
## 5 P0537 Slowed-down 531
## 6 P0899 Normal 433
## 7 P0491 Slowed-down 1274
## 8 P0122 Slowed-down 3146
## 9 P0661 Slowed-down 626
## 10 P0805 Slowed-down 1075
## 11 P0070 Slowed-down 390
## 12 P0379 Slowed-down 302
## 13 P0201 Slowed-down 2561
## 14 P0787 Normal 888
## 15 P1010 Slowed-down 2322
## 16 P1177 Slowed-down 391
## 17 P1118 Slowed-down 719
## 18 P0929 Slowed-down 820
## 19 P1071 Slowed-down 352An additional 38 betting times (of which 28 are in the slowed down version) within the range of 150 to 300 seconds are not shown here.
We can also look at some overall statistics of the betting times:
## # A tibble: 2 × 5
## exp_cond time_mean time_median time_sd time_IQR
## <fct> <dbl> <dbl> <dbl> <dbl>
## 1 Normal 20.95 15 25.73 10
## 2 Slowed-down 88.63 68 144.6 15Distribution of Proportion Bet
Our main DV clearly does not look normally distributed.
## # A tibble: 1 × 3
## gamble_at_all gamble_everything proportion_bet_rest
## <dbl> <dbl> <dbl>
## 1 0.808 0.144 0.399Binomial confidence or credibility intervals for the probability to gamble at all:
## method x n mean lower upper
## 1 agresti-coull 810 1002 0.8083832 0.7828261 0.8315849
## 2 asymptotic 810 1002 0.8083832 0.7840141 0.8327523
## 3 bayes 810 1002 0.8080758 0.7835673 0.8322395
## 4 cloglog 810 1002 0.8083832 0.7826222 0.8314267
## 5 exact 810 1002 0.8083832 0.7826229 0.8323168
## 6 logit 810 1002 0.8083832 0.7828269 0.8315792
## 7 probit 810 1002 0.8083832 0.7830742 0.8317968
## 8 profile 810 1002 0.8083832 0.7832438 0.8319480
## 9 lrt 810 1002 0.8083832 0.7832510 0.8319447
## 10 prop.test 810 1002 0.8083832 0.7823333 0.8320295
## 11 wilson 810 1002 0.8083832 0.7828544 0.8315565Numbers of Spins
The l0 variables refer to the subset of participants who made at least 1 spin (i.e., number of spins is larger than zero).
## # A tibble: 1 × 6
## mean median sd mean_l0 median_l10 sd_l0
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 4.7 2 9.2 5.8 3 10## # A tibble: 2 × 5
## exp_cond mean sd mean_l0 sd_l0
## <fct> <dbl> <dbl> <dbl> <dbl>
## 1 Normal 5.96 11.7 7.39 12.6
## 2 Slowed-down 3.41 5.66 4.20 6.02The following shows the top 10 largest bet counts (n is the number of participants with this bet count):
## # A tibble: 69 × 3
## # Groups: exp_cond [2]
## exp_cond bet_count n
## <fct> <dbl> <int>
## 1 Normal 136 1
## 2 Normal 130 1
## 3 Normal 81 1
## 4 Slowed-down 62 1
## 5 Slowed-down 53 1
## 6 Normal 51 1
## 7 Normal 49 1
## 8 Normal 47 1
## 9 Normal 46 1
## 10 Normal 42 1
## # … with 59 more rowsProportion of participants with that many spins:
## # A tibble: 2 × 6
## exp_cond only1 morethan1 upto2 only2 morethan2
## <fct> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Normal 0.211 0.789 0.355 0.144 0.645
## 2 Slowed-down 0.278 0.722 0.536 0.258 0.464Average bet size
The following table shows the average bet sizes across all bets (and not across participants):
## # A tibble: 1 × 3
## mean sd n
## <dbl> <dbl> <int>
## 1 0.727 0.628 4688## # A tibble: 2 × 4
## exp_cond mean sd n
## <fct> <dbl> <dbl> <int>
## 1 Normal 0.667 0.608 2978
## 2 Slowed-down 0.832 0.648 1710The following tables show the averages of the participant average bet sizes (i.e., averages of averages):
## # A tibble: 1 × 3
## mean sd n
## <dbl> <dbl> <int>
## 1 0.904 0.602 1002## # A tibble: 2 × 4
## exp_cond mean sd n
## <fct> <dbl> <dbl> <int>
## 1 Normal 0.870 0.593 500
## 2 Slowed-down 0.937 0.610 502Model 1: Zero-One Inflated Beta Regression on Proportion Bet
We use a custom parameterization of a zero-one-inflated beta-regression model (see also here). The likelihood of the model is given by:
\[\begin{align} f(y) &= (1 - g) & & \text{if } y = 0 \\ f(y) &= g \times e & & \text{if } y = 1 \\ f(y) &= g \times (1 - e) \times \text{Beta}(a,b) & & \text{if } y \notin \{0, 1\} \\ a &= \mu \times \phi \\ b &= (1-\mu) \times \phi \end{align}\]
Where \(1 - g\) is the zero inflation probability, zipp is \(g\) and reflects the probability to gamble, \(e\) is the conditional one-inflation probability (coi) or conditional probability to gamble everything (i.e., conditional probability to have a value of one, if one gambles), \(\mu\) is the mean of the beta distribution (Intercept), and \(\phi\) is the precision of the beta distribution (phi). As we use Stan for modelling, we need to model on the real line and need appropriate link functions. For \phi the link is log (inverse is exp()), for all other parameters it is logit (inverse is plogis()).
We fit this model and add experimental condition as a factor to the three main model parameters (i.e., only the precision parameter is fixed across conditions). The following table provides the overview of the model and all model parameters and show good convergence.
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ exp_cond
## phi ~ 1
## zipp ~ exp_cond
## coi ~ exp_cond
## Data: part2 (Number of observations: 1002)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.32 0.05 -0.42 -0.22 1.00 141787 77417
## phi_Intercept 1.28 0.05 1.18 1.37 1.00 137922 79202
## zipp_Intercept 1.42 0.11 1.21 1.65 1.00 140249 75297
## coi_Intercept -1.61 0.13 -1.88 -1.36 1.00 148706 76108
## exp_condSlowedMdown -0.17 0.07 -0.30 -0.03 1.00 144406 78328
## zipp_exp_condSlowedMdown 0.03 0.16 -0.29 0.35 1.00 142657 79793
## coi_exp_condSlowedMdown -0.35 0.20 -0.75 0.04 1.00 144473 79866
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the normal speed condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the slowed down condition from the normal speed condition. These differences are given on the logit scale.
The model does not have any obvious problems, even without priors for the condition specific effects.
Posterior Predictive Checks
As expected the synthetic data generated from the model looks a lot like the actual data. This suggests that the model is adequate for the data.
Model Estimates
We first give the table showing the posterior means and 95% CIs.
## # A tibble: 6 × 8
## parameter condition estimate .lower .upper .width .point .interval
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Gamble at all? Normal 0.805 0.770 0.839 0.95 mean qi
## 2 Gamble at all? Slowed-down 0.810 0.774 0.843 0.95 mean qi
## 3 Gamble everything? Normal 0.167 0.132 0.205 0.95 mean qi
## 4 Gamble everything? Slowed-down 0.124 0.0936 0.157 0.95 mean qi
## 5 Proportion bet? Normal 0.421 0.397 0.445 0.95 mean qi
## 6 Proportion bet? Slowed-down 0.381 0.358 0.404 0.95 mean qiFor the zero-one inflated components, we can compare the model estimates with the data. Not unsurprisingly, they match quite well.
## # A tibble: 2 × 3
## exp_cond gamble_at_all gamble_everything
## <fct> <dbl> <dbl>
## 1 Normal 0.806 0.166
## 2 Slowed-down 0.811 0.123The following is the main results figure on the level of condition.
Difference distribution
We can also focus and look at the difference distributions.
## # A tibble: 3 × 8
## parameter condition estimate .lower .upper .width .point .interval
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Gamble at all? " " 0.00468 -0.0444 0.0537 0.95 mean qi
## 2 Gamble everything? " " -0.0433 -0.0921 0.00490 0.95 mean qi
## 3 Proportion bet? " " -0.0399 -0.0727 -0.00705 0.95 mean qiSame as a figure.
Difference Distributions with Possible Priors
The following plot shows the estimated difference of the regular from the slowed down condition with overlaid density estimate (in black) and some possible prior distributions in colour (note again that the model did not actually include any priors). These differences are shown on the linear scale before applying the logistic link function. These priors are normal priors (who have a higher peak at 0 compared to Cauchy and t) with different SDs.
In this figure we considered three prior width. For a prior width of SD = 0.25 we expect with 95% that the largest possible effect we observe is 12.01% on the response scale. For a prior width of SD = 0.5 we expect with 95% that the largest possible effect we observe is 22.71% on the response scale. For a prior width of SD = 0.5 we expect with 95% that the largest possible effect we observe is 37.65% on the response scale.
Combined Estimates
## # A tibble: 2 × 7
## condition overall .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Normal 0.417 0.387 0.446 0.95 mean qi
## 2 Slowed-down 0.371 0.343 0.399 0.95 mean qi## # A tibble: 1 × 7
## condition overall .lower .upper .width .point .interval
## <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Slowed-down - Normal -0.0462 -0.0865 -0.00578 0.95 mean qi
Individual Model Estimates
Model 1 With PGSI
The covariate scores do not differ between conditions (evidence for the null).
## Bayes factor analysis
## --------------
## [1] Intercept only : 12.59012 ±0.02%
##
## Against denominator:
## pgsi ~ exp_cond
## ---
## Bayes factor type: BFlinearModel, JZSThe following table shows mean and SDs of the covariates per group.
## # A tibble: 2 × 3
## exp_cond pgsi_mean pgsi_sd
## <fct> <dbl> <dbl>
## 1 Normal 3.03 3.89
## 2 Slowed-down 3.15 3.91Model with Main Effects
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ exp_cond + pgsi_c
## phi ~ 1
## zipp ~ exp_cond + pgsi_c
## coi ~ exp_cond + pgsi_c
## Data: part2 (Number of observations: 1002)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.32 0.05 -0.42 -0.22 1.00 196171 74746
## phi_Intercept 1.28 0.05 1.19 1.38 1.00 190455 75256
## zipp_Intercept 1.45 0.11 1.23 1.68 1.00 197926 75760
## coi_Intercept -1.65 0.14 -1.92 -1.38 1.00 193801 77931
## exp_condSlowedMdown -0.17 0.07 -0.31 -0.03 1.00 200255 72111
## pgsi_c 0.02 0.01 0.01 0.04 1.00 207125 76876
## zipp_exp_condSlowedMdown 0.02 0.16 -0.29 0.34 1.00 196720 75198
## zipp_pgsi_c 0.06 0.02 0.02 0.11 1.00 190564 79574
## coi_exp_condSlowedMdown -0.38 0.20 -0.78 0.02 1.00 193066 78477
## coi_pgsi_c 0.08 0.02 0.03 0.12 1.00 191732 81004
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the normal speed condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the slowed down condition from the normal speed condition. These differences are given on the logit scale.
## # A tibble: 3 × 8
## parameter condition estimate .lower .upper .width .point .interval
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Gamble at all? " " 0.00368 -0.0446 0.0519 0.95 mean qi
## 2 Gamble everything? " " -0.0449 -0.0923 0.00231 0.95 mean qi
## 3 Proportion bet? " " -0.0412 -0.0743 -0.00802 0.95 mean qi
Plot of Relationships and BANOVA
The figure below provides an alternative visualisation of the relationships between covariates and betting behaviour. In particular, participants were categorized into one of three experimental betting behavior groups: participants who did not bet at all (“none”, 14% of participants), participants who bet some of their money (68% of participants), and participants who bet “all” of their money (18%). For both gambling scales we see a positive relationship between the betting behavior group and the gambling score. Participants who bet more have on average higher scores on the two gambling scales.
##
## none some all
## 0.1916168 0.6916168 0.1167665
This is supported by Bayesian ANOVAs with Bayes factors of over 100 for the effect of betting behavior group on the PGSI scores. However, this effect was not moderated by gambling speed condition. In particular, there was evidence for the absence of both a main effect of gambling message condition and an interaction of gambling message condition with betting behaviour group for both gambling scale scores (Bayes factors for the null > 25).
## Bayes factor analysis
## --------------
## [1] exp_cond : 0.07942733 ±0.02%
## [2] gamble_cat : 146.7407 ±0.03%
## [3] exp_cond + gamble_cat : 12.69162 ±1.45%
## [4] exp_cond + gamble_cat + exp_cond:gamble_cat : 0.6253256 ±1.88%
##
## Against denominator:
## Intercept only
## ---
## Bayes factor type: BFlinearModel, JZS## denominator
## numerator exp_cond gamble_cat exp_cond + gamble_cat exp_cond + gamble_cat + exp_cond:gamble_cat
## gamble_cat 1847.483 1 11.56201 234.6628Model With Interactions
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ exp_cond * pgsi_c
## phi ~ 1
## zipp ~ exp_cond * pgsi_c
## coi ~ exp_cond * pgsi_c
## Data: part2 (Number of observations: 1002)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.32 0.05 -0.42 -0.22 1.00 137427 77963
## phi_Intercept 1.28 0.05 1.19 1.38 1.00 133519 74471
## zipp_Intercept 1.44 0.12 1.22 1.67 1.00 130184 74845
## coi_Intercept -1.64 0.14 -1.91 -1.38 1.00 151293 75658
## exp_condSlowedMdown -0.17 0.07 -0.31 -0.03 1.00 143580 76086
## pgsi_c 0.03 0.01 -0.00 0.05 1.00 92207 79088
## exp_condSlowedMdown:pgsi_c -0.01 0.02 -0.04 0.03 1.00 94198 77196
## zipp_exp_condSlowedMdown 0.04 0.17 -0.28 0.37 1.00 122384 78161
## zipp_pgsi_c 0.05 0.03 -0.01 0.12 1.00 95058 76013
## zipp_exp_condSlowedMdown:pgsi_c 0.04 0.05 -0.06 0.13 1.00 92926 77410
## coi_exp_condSlowedMdown -0.40 0.21 -0.81 0.01 1.00 124272 79569
## coi_pgsi_c 0.07 0.03 0.01 0.13 1.00 89959 77242
## coi_exp_condSlowedMdown:pgsi_c 0.02 0.04 -0.07 0.11 1.00 88532 76590
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the normal speed condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the slowed down condition from the normal speed condition. These differences are given on the logit scale.
## # A tibble: 3 × 8
## parameter condition estimate .lower .upper .width .point .interval
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Gamble at all? " " 0.00671 -0.0428 0.0563 0.95 mean qi
## 2 Gamble everything? " " -0.0469 -0.0954 0.00143 0.95 mean qi
## 3 Proportion bet? " " -0.0413 -0.0743 -0.00811 0.95 mean qi
Model 2: Total Number of Spins
## Family: negbinomial
## Links: mu = log; shape = identity
## Formula: bet_count | trunc(lb = 1) ~ exp_cond
## Data: part_nozero (Number of observations: 810)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.02 0.34 0.27 1.44 1.00 21689 15312
## exp_condSlowedMdown -0.80 0.13 -1.05 -0.56 1.00 43585 44105
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 0.18 0.06 0.07 0.29 1.00 21194 14580
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
The data seems to be well described by the model.
When we zoom in (i.e., ignore data points above 50 for the plot), we can see the that the real and synthetic data match quite well.
## # A tibble: 2 × 7
## exp_cond mean .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Normal 2.93 1.31 4.23 0.95 median qi
## 2 Slowed-down 1.32 0.582 1.94 0.95 median qi
## # A tibble: 1 × 7
## exp_cond mean .lower .upper .width .point .interval
## <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Slowed-down - Normal -1.58 -2.52 -0.681 0.95 median qi## # A tibble: 1 × 2
## exp_cond prop_larger_0
## <chr> <dbl>
## 1 Slowed-down - Normal 0Model 2 With PGSI
With Main Effects
## Family: negbinomial
## Links: mu = log; shape = identity
## Formula: bet_count | trunc(lb = 1) ~ exp_cond + pgsi_c
## Data: part_nozero (Number of observations: 810)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.01 0.36 0.26 1.44 1.00 30337 18078
## exp_condSlowedMdown -0.80 0.13 -1.05 -0.56 1.00 62314 53591
## pgsi_c 0.00 0.02 -0.03 0.03 1.00 68753 54882
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 0.17 0.06 0.06 0.29 1.00 29392 17756
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
## # A tibble: 2 × 7
## exp_cond mean .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Normal 2.91 1.30 4.21 0.95 median qi
## 2 Slowed-down 1.31 0.569 1.93 0.95 median qi
With Interactions
## Family: negbinomial
## Links: mu = log; shape = identity
## Formula: bet_count | trunc(lb = 1) ~ exp_cond * pgsi_c
## Data: part_nozero (Number of observations: 810)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 1.00 0.36 0.23 1.43 1.00 31512 21588
## exp_condSlowedMdown -0.80 0.13 -1.06 -0.56 1.00 64418 56874
## pgsi_c 0.00 0.02 -0.04 0.05 1.00 48764 53707
## exp_condSlowedMdown:pgsi_c -0.01 0.03 -0.07 0.06 1.00 48398 56081
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 0.17 0.06 0.06 0.29 1.00 30894 21196
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
## NOTE: Results may be misleading due to involvement in interactions
## NOTE: Results may be misleading due to involvement in interactions## # A tibble: 2 × 7
## exp_cond mean .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Normal 2.88 1.26 4.19 0.95 median qi
## 2 Slowed-down 1.29 0.545 1.92 0.95 median qi## NOTE: Results may be misleading due to involvement in interactions
Model 3: Average Bet Size
The following shows the distribution of average bet sizes for those that bet at least once. The left plot shows the actual averages and the right plot the average scaled to the 0 to 1 range (i.e., after dividing by 2 as pre-registered). We can see that the distribution of actual averages is multi-modal with peaks at several prominent numbers: the first peak at the minimum possible bet size 0.1, the second peak at 0.5, the largest peak at the center at 1 and finally a large peak at the maximum of 2. As pre-registered we will analyse the data with a one-inflated beta-regression model.
## Family: oib
## Links: mu = logit; phi = log; oi = logit
## Formula: average_scaled ~ exp_cond
## phi ~ 1
## oi ~ exp_cond
## Data: part_nozero (Number of observations: 810)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.52 0.05 -0.62 -0.42 1.00 117234 81473
## phi_Intercept 1.23 0.05 1.14 1.33 1.00 118483 80530
## oi_Intercept -2.05 0.16 -2.37 -1.75 1.00 115208 76360
## exp_condSlowedMdown 0.06 0.07 -0.08 0.20 1.00 118975 79362
## oi_exp_condSlowedMdown 0.21 0.21 -0.20 0.63 1.00 116127 78606
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
When comparing the actual data with synthetic data from the model, we can see an obvious problem due to the use of prominent numbers. Whereas the peak at 1 is nicely recovered by the model, the other peaks are not. However, the overall shape of the synthetic data is at least similar to the shape of the observed data.
## Joining, by = c("exp_cond", ".chain", ".iteration", ".draw")
## # A tibble: 4 × 8
## parameter exp_cond value .lower .upper .width .point .interval
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Mean bet size Normal 0.373 0.350 0.396 0.95 median qi
## 2 Mean bet size Slowed-down 0.387 0.363 0.410 0.95 median qi
## 3 Bet maximum Normal 0.114 0.0856 0.148 0.95 median qi
## 4 Bet maximum Slowed-down 0.138 0.107 0.173 0.95 median qiDifference distribution:
## # A tibble: 2 × 8
## parameter exp_cond value .lower .upper .width .point .interval
## <fct> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Mean bet size Slowed-down - Normal 0.0140 -0.0187 0.0465 0.95 median qi
## 2 Bet maximum Slowed-down - Normal 0.0235 -0.0222 0.0690 0.95 median qi
We can also re-transforming the model estimates to the £-scale. This gives the following two numbers:
## # A tibble: 2 × 7
## exp_cond pred_mean .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Normal 0.889 0.834 0.947 0.95 median qi
## 2 Slowed-down 0.943 0.886 1.00 0.95 median qiOf course, we can also look at the corresponding difference distribution:
## # A tibble: 1 × 7
## exp_cond pred_mean .lower .upper .width .point .interval
## <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Slowed-down - Normal 0.0538 -0.0269 0.133 0.95 median qi
Model 3 with PGSI
With Main Effect
## Family: oib
## Links: mu = logit; phi = log; oi = logit
## Formula: average_scaled ~ exp_cond + pgsi_c
## phi ~ 1
## oi ~ exp_cond
## Data: part_nozero (Number of observations: 810)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.52 0.05 -0.62 -0.43 1.00 132096 80699
## phi_Intercept 1.25 0.05 1.16 1.35 1.00 131280 79940
## oi_Intercept -2.05 0.16 -2.37 -1.75 1.00 131684 72906
## exp_condSlowedMdown 0.06 0.07 -0.08 0.20 1.00 133382 77673
## pgsi_c 0.03 0.01 0.01 0.05 1.00 135762 77110
## oi_exp_condSlowedMdown 0.21 0.21 -0.20 0.63 1.00 133556 77427
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
## Joining, by = c("exp_cond", ".chain", ".iteration", ".draw")
Difference distribution:
With Interaction
## Family: oib
## Links: mu = logit; phi = log; oi = logit
## Formula: average_scaled ~ exp_cond * pgsi_c
## phi ~ 1
## oi ~ exp_cond
## Data: part_nozero (Number of observations: 810)
## Draws: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup draws = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.52 0.05 -0.62 -0.43 1.00 158731 77698
## phi_Intercept 1.25 0.05 1.16 1.35 1.00 161915 76108
## oi_Intercept -2.05 0.16 -2.36 -1.75 1.00 144370 74781
## exp_condSlowedMdown 0.06 0.07 -0.08 0.19 1.00 161101 72690
## pgsi_c 0.02 0.01 -0.01 0.04 1.00 105929 81572
## exp_condSlowedMdown:pgsi_c 0.03 0.02 -0.01 0.06 1.00 105092 81066
## oi_exp_condSlowedMdown 0.21 0.21 -0.21 0.63 1.00 155452 74101
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Difference distribution:
Riskiness of Bets
Do either of the spin speeds affect the riskiness of bets chosen in the roulette? Different bets come with different potential payoffs in roulette. If £1 is bet, then this can provide a total payoff of between £2 (e.g., bets on red or black) and £36 (bets on a single number). These numbers, 2 and 36, are the decimal odds for these two bets, representing the total potential payoff Additionally, gamblers can place multiple bets per spin, (e.g., betting £0.50 on red and £0.50 on 7). Since the number 7 is a red colour, the bet on 7 can only win if the first bet on red also wins. Multiple bets per spin can either be placed in a way that accentuates risk, as in this example, or in a way that hedges risk (for example, a bet on black added to the bet on 7, or betting on reds and blacks together). The purpose of this exploratory analysis is to see if either warning label affects risk taking.
Following our pre-registration we have measures the amount of risk taken by looking at the variance of the decimal odds for each number in the roulette table. It will measure the concentration of the bet, with more risk represented by more concentrated bets (e.g., betting on a single number), and lower risk represented by more spread bets (e.g., betting on reds, betting on evens). For example, if a participant places a bet on number 7, the decimal odds for that number is 36 (36 times the amount bet if the roulette stops on the number 7). If a participant places a bet on red, then every red number on the table gets a decimal odd of 2 (the participant will win twice the amount bet if the roulette stops on any red number). In order to calculate the proposed risk variable, we will also assign decimal odds of zero for the numbers in which participants did not bet (since they win zero times the amount bet if the roulette stops on those numbers). In the two examples above, every number except 7 will be assigned zero, and every non-red number will be assigned zero, respectively. The risk variable will be the variance of the array of decimal odds for every number on the roulette table, taking into account the bets the participant has placed, and including zeroes for non-winning numbers. The value of this risk variable can range between 0.03 and 35.03. The value cannot be zero in our task because participants are not able to bet the same amount across all numbers including the zero due to the limitations in the values of the available tokens and total bet amounts. Higher numbers indicate more risk taking (with the highest, 35.03, associated with concentrating the bet on a single number). Lower numbers indicate lower risk taking (with the lowest, 0.03, associated with spreading the bet across every red and black number, excluding zero).
The following plot shows the distribution of this variable after aggregating within participants. The left plot shows the original variable on the scale from 0.03 to 35.03. The middle plot shows the variable after dividing by 36 so that the variable ranges from just above 0 to just below 1. The right plot shows the variable after subtracting 0.03 and dividing by 35 so that the smallest value is mapped to 0 and the largest value is mapped to 1.
Beta Regression
We first analyse the data with a beta-regression model. For this, the data needs to be between 0 and 1, but exclude exactly 0 and 1. Consequently, we use the transformation shown above in the middle panel. As before, we fit the data and allow for an effect of gambling message.
The model does not show any obvious problems. In addition, we can see that the 95%-CIs for the slowed down effect includes 0.
## Family: beta
## Links: mu = logit; phi = log
## Formula: average_odds2 ~ exp_cond
## phi ~ 1
## Data: bets_av (Number of observations: 810)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -2.06 0.06 -2.17 -1.94 1.00 2493 2417
## phi_Intercept 1.59 0.06 1.48 1.69 1.00 2574 2729
## exp_condSlowedMdown -0.04 0.07 -0.18 0.09 1.00 2802 2690
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
The posterior predictive distribution shows some problems, but at least the shape of the synthetic data is similar to the distribution of the actual data (posterior predictive distributions of a model assuming a Gaussian response distribution was considerably worse and is therefore not included here).
The following tables show the average riskiness per condition and differences from the no warning message group on the fitted scale, i.e., the (0, 1) scale used for the beta regression.
## # A tibble: 2 × 7
## condition estimate .lower .upper .width .point .interval
## <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Normal 0.114 0.103 0.125 0.95 mean qi
## 2 SlowedDown 0.110 0.0990 0.121 0.95 mean qi## # A tibble: 1 × 7
## condition estimate .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 SlowedDown - Normal -0.00402 -0.0178 0.00894 0.95 mean qiSession Info (R and Package Numbers)
## R version 4.1.0 (2021-05-18)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.5 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
##
## locale:
## [1] LC_CTYPE=en_GB.UTF-8 LC_NUMERIC=C LC_TIME=en_GB.UTF-8
## [4] LC_COLLATE=en_GB.UTF-8 LC_MONETARY=en_GB.UTF-8 LC_MESSAGES=en_GB.UTF-8
## [7] LC_PAPER=en_GB.UTF-8 LC_NAME=C LC_ADDRESS=C
## [10] LC_TELEPHONE=C LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] emmeans_1.7.0 binom_1.1-1 BayesFactor_0.9.12-4.2 Matrix_1.3-3
## [5] coda_0.19-4 tidybayes_3.0.1 brms_2.16.1 Rcpp_1.0.7
## [9] forcats_0.5.1 stringr_1.4.0 dplyr_1.0.7 purrr_0.3.4
## [13] readr_2.0.2 tidyr_1.1.4 tibble_3.1.5 ggplot2_3.3.5
## [17] tidyverse_1.3.1 checkpoint_1.0.0
##
## loaded via a namespace (and not attached):
## [1] readxl_1.3.1 backports_1.2.1 plyr_1.8.6 igraph_1.2.7
## [5] splines_4.1.0 svUnit_1.0.6 crosstalk_1.1.1 rstantools_2.1.1
## [9] inline_0.3.19 digest_0.6.28 htmltools_0.5.2 rsconnect_0.8.24
## [13] fansi_0.5.0 magrittr_2.0.1 checkmate_2.0.0 tzdb_0.1.2
## [17] modelr_0.1.8 RcppParallel_5.1.4 matrixStats_0.61.0 vroom_1.5.5
## [21] xts_0.12.1 prettyunits_1.1.1 colorspace_2.0-2 rvest_1.0.2
## [25] ggdist_3.0.0 haven_2.4.3 xfun_0.27 callr_3.7.0
## [29] crayon_1.4.1 jsonlite_1.7.2 lme4_1.1-27.1 zoo_1.8-9
## [33] glue_1.4.2 gtable_0.3.0 MatrixModels_0.5-0 V8_3.4.2
## [37] distributional_0.2.2 pkgbuild_1.2.0 rstan_2.21.2 abind_1.4-5
## [41] scales_1.1.1 mvtnorm_1.1-3 DBI_1.1.1 miniUI_0.1.1.1
## [45] xtable_1.8-4 bit_4.0.4 stats4_4.1.0 StanHeaders_2.21.0-7
## [49] DT_0.19 htmlwidgets_1.5.4 httr_1.4.2 threejs_0.3.3
## [53] arrayhelpers_1.1-0 posterior_1.1.0 ellipsis_0.3.2 pkgconfig_2.0.3
## [57] loo_2.4.1 farver_2.1.0 sass_0.4.0 dbplyr_2.1.1
## [61] utf8_1.2.2 labeling_0.4.2 tidyselect_1.1.1 rlang_0.4.12
## [65] reshape2_1.4.4 later_1.3.0 munsell_0.5.0 cellranger_1.1.0
## [69] tools_4.1.0 cli_3.0.1 generics_0.1.0 broom_0.7.9
## [73] ggridges_0.5.3 evaluate_0.14 fastmap_1.1.0 yaml_2.2.1
## [77] bit64_4.0.5 processx_3.5.2 knitr_1.36 fs_1.5.0
## [81] pbapply_1.5-0 nlme_3.1-152 mime_0.12 projpred_2.0.2
## [85] xml2_1.3.2 compiler_4.1.0 bayesplot_1.8.1 shinythemes_1.2.0
## [89] rstudioapi_0.13 gamm4_0.2-6 curl_4.3.2 reprex_2.0.1
## [93] bslib_0.3.1 stringi_1.7.5 highr_0.9 ps_1.6.0
## [97] Brobdingnag_1.2-6 lattice_0.20-44 nloptr_1.2.2.2 markdown_1.1
## [101] ggsci_2.9 shinyjs_2.0.0 tensorA_0.36.2 vctrs_0.3.8
## [105] pillar_1.6.4 lifecycle_1.0.1 jquerylib_0.1.4 bridgesampling_1.1-2
## [109] estimability_1.3 cowplot_1.1.1 httpuv_1.6.3 R6_2.5.1
## [113] promises_1.2.0.1 gridExtra_2.3 codetools_0.2-18 boot_1.3-28
## [117] colourpicker_1.1.1 MASS_7.3-54 gtools_3.9.2 assertthat_0.2.1
## [121] withr_2.4.2 shinystan_2.5.0 mgcv_1.8-35 parallel_4.1.0
## [125] hms_1.1.1 grid_4.1.0 minqa_1.2.4 rmarkdown_2.11
## [129] shiny_1.7.1 lubridate_1.8.0 base64enc_0.1-3 dygraphs_1.1.1.6